Theorem sseqtrrd 3218

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrrd.1	|- ( ph -> A C_ B )
sseqtrrd.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtrrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2199	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3217	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  sseqtrrid  3230  fnfvima  5793  tfrlemiubacc  6383  tfr1onlemubacc  6399  tfrcllemubacc  6412  rdgivallem  6434  nnnninf  7185  nninfwlpoimlemg  7234  dfphi2  12358  ctinf  12587  imasaddfnlemg  12897  imasaddvallemg  12898  subsubm  13055  subsubg  13267  subsubrng  13710  subsubrg  13741  lidlss  13972  toponss  14194  ssntr  14290  iscnp3  14371  cnprcl2k  14374  tgcn  14376  tgcnp  14377  ssidcn  14378  cncnp  14398  txcnp  14439  imasnopn  14467  hmeontr  14481  blssec  14606  blssopn  14653  xmettx  14678  metcnp  14680  plyaddlem1  14893  plymullem1  14894  nnsf  15495  nninfsellemsuc  15502